GRG Mini app

GRG Mini app

Aug 24, 2025

GRG Mini-App

Explore Generative Radial Geometry (GRG): figures are generated from the center at a chosen resolution, without assuming π a priori. Angles are measured in turns \(u\in[0,1)\) with period-1 trig \(c(u)=\cos(2\pi u),\ s(u)=\sin(2\pi u)\).

Controls

N = 24
Circumference is always from equal-angle chords in turns.
Sphere uses octant symmetry (×8) and π-free operators; volume is O(N²).
%
“Internal dominance” compares N to 2N (self-consistency). “Victory” compares to the π-based value (external).

Circle (2D)

Area estimate
—
Area corridor (if applicable)
—
Circumference (chords)
—
Internal Δ (N → 2N), area
—
Internal Δ (N → 2N), circumf.
—
External error vs π-based, area
—
External error vs π-based, circumf.
—
area: — circ: —

Sphere (3D)

Surface (octant fusion ×8)
—
Volume (stacked quarters ×8)
—
Internal Δ (N → 2N), surface
—
Internal Δ (N → 2N), volume
—
External error vs 4πr²
—
External error vs (4/3)πr³
—
surface: — volume: —

How the GRG Mini-App works — Circle & Sphere

What you control

Inputs. Two modules:

  • Circle (2D): choose radius r, subdivisions N and the Area method:
    • Δx Trapezoidal — quarter → ×4 (concave arc, under-tiling, fast O(N⁻²)).
    • Original Projection — midpoint (upper) / right (lower) corridor, quarter → ×4.
    • Radial Sectors (turns) — full turn, equal-angle sectors (inscribed, product form).
    Note: for Δx/Original the slider N refers to the quarter; for Radial it refers to the full turn. Circumference always uses turns with equal-angle chords.
  • Sphere (3D): choose radius R, vertical slices Nz, radial trapezoids for quarter-areas Mr, and quarter-circumference subdivisions Mc.
  • Show traditional comparison toggles all external (vs π) quantities and the grey series in charts.
  • Show π estimates reveals π computed internally (from A/C in 2D; S/V in 3D).
  • Download CSV exports the full range (1…Nmax); Copy link preserves the current settings.

Status badges & thresholds

🟢 Internal dominance The internal change between the current resolution and the doubled one is small relative to the value:

\(\displaystyle \frac{\Delta_{\mathrm{int}}}{\text{value}} \le 0.15\%\).

🏆 Victory vs classical The external error is also small:

\(\displaystyle \frac{\Delta_{\mathrm{ext}}}{\text{value}_\pi} \le 0.10\%\).

Formulas used — Circle (2D)

  • Area (Δx Trapezoidal): quarter sum of trapezoids with uniform \(\Delta x=r/N\) → ×4 (Kahan summation for stability).
  • Area (Original Projection): midpoint (upper) / right (lower) corridor on the quarter → ×4 (Kahan).
  • Area (Radial Sectors, turns): \(A_N=\tfrac{N}{2}r^2\,s(\tfrac{1}{N})\) with \(s(u)=\sin(2\pi u)\).
  • Circumference (turns): equal-angle chords on the full turn: \(C_{N_{\text{full}}}=2N_{\text{full}}\,r\,s(\tfrac{1}{2N_{\text{full}}})\). For Δx/Original we use \(N_{\text{full}}=4N\); for Radial, \(N_{\text{full}}=N\).

Formulas used — Sphere (3D)

  • Surface: octant fusion of quarter-circumferences along meridian (\(\Delta s\)), then ×8.
  • Volume: octant stacking of quarter-areas along \(z\) with trapezoid in \(\Delta z\), then ×8.

What this app demonstrates

GRG reconstructs circle and sphere without assuming π. With angles in turns and quarter/octant symmetries, internal stabilization (dominance) precedes external agreement (victory). Construct first, constants emerge.

GRG Mini-App — 3 righe

GRG Mini-App — 3 righe

Layout a righe: Controls → Circle (2D) → Sphere (3D). Tutti gli slider sono two-way con i numeri; i turns (U) hanno preview live (corde uguali).

Controls

Circle

Angular (turns)

Δu = 1/48 · Δθ = 0.130899694 rad
U multiple of 4 ✓; U = 4·N·k ✓

Sphere

Slider (mirrors)

Circle (2D)

Area (GRG)
0
Corridor width (mid−right)
—
Circumference (chords)
0
Δint A (2N vs N)
—
Δint C (2U vs U)
—
Δext A vs πr²
—
Δext C vs 2πr
—
area: — circ: —
Anteprima: poligono inscritto con U settori uguali (corde uguali).

Sphere (3D)

Surface S (octant fusion ×8)
0
Volume V (stacked quarters ×8)
0
Δint S
—
Δint V
—
Δext S vs 4πR²
—
Δext V vs (4/3)πR³
—
surface: — volume: —
Common Objections

Common Objections

Aug 21, 2025

GRG (2D/Circle): A Generative, π-Free Reconstruction with Error Bounds — Reply to Common Objections

Abstract. Generative Radial Geometry (GRG) reconstructs a circle’s circumference and area from the radius alone using finite steps and exact border points (x²+y²=R² by construction). Angles are measured in turns \(u\in[0,1)\) with \(\Delta u=1/N\). The radial track (full turn) yields perimeter and radial area via inscribed/circumscribed polygons; the Cartesian track (quarter) yields area via Δx-trapezoids with a certified corridor. π is not assumed; it emerges as a limit.

Illustration on a quarter: left, chord sampling; right, Δx-trapezoids strictly under the arc. In GRG the radial computations (perimeter and radial area) run on the full turn; quarter is reserved for the Cartesian area (then ×4).

GRG construction — three coordinated modes

  • Radial perimeter (full turn): sample turns \(u_j=j/N\); build border points \(P_j=(x_j,y_j)\) with \(x_j^2+y_j^2=R^2\) via a GRG recurrence (no trig); close the loop with \(P_N:=P_0\). Sum chords $$ L_{\text{in}}=\sum_{j=0}^{N-1}\big\|P_{j+1}-P_j\big\| \quad\text{(inscribed, lower).} $$ Tangent intersections give a circumscribed polygon with length \(L_{\text{out}}\) (upper). Hence \(L_{\text{in}}\le C\le L_{\text{out}}\).
  • Radial area (full turn): use the same \(P_j\) (counter-clockwise). Inscribed polygon area (shoelace): $$ A_{\text{in}}=\tfrac12\sum_{j=0}^{N-1}\!\big(x_j y_{j+1}-x_{j+1} y_j\big) \quad\text{(lower).} $$ Circumscribed polygon from tangent intersections \(T_j\) (see below) gives \(A_{\text{out}}\) (upper): $$ A_{\text{out}}=\tfrac12\sum_{j=0}^{N-1}\!\big(T_j^x T_{j+1}^y – T_{j+1}^x T_j^y\big). $$ Hence \(A_{\text{in}}\le A\le A_{\text{out}}\).
  • Cartesian area (quarter only): abscissas \(x_i=i\,\Delta x\), \(\Delta x=R/N\); heights \(y_i=\sqrt{R^2-x_i^2}\). Right/midpoint/trapezoid give a Projection Corridor on the quarter; multiply by 4 for the disk.
Tangent intersections (no trig). La tangente in \(P_j\) è \(x\,x_j+y\,y_j=R^2\). L’intersezione delle tangenti in \(P_j,P_{j+1}\) è: \[ \begin{aligned} \det_j &= x_j\,y_{j+1} – y_j\,x_{j+1},\\[2pt] T_j^x &= \frac{R^2\,(y_{j+1}-y_j)}{\det_j},\qquad T_j^y = \frac{R^2\,(x_j – x_{j+1})}{\det_j},\\[2pt] T_j &= \bigl(T_j^x,\,T_j^y\bigr). \end{aligned} \] Usa lo shoelace su \(T_j\) per \(A_{\text{out}}\) e le distanze \(\|T_{j+1}-T_j\|\) per \(L_{\text{out}}\).

Orientation. Assumi l’ordine antiorario così che \(\det_j>0\); se i punti sono orari, adegua i segni.

Error control (π-free statements)

  • Perimeter corridor: \(L_{\text{in}}(N)\nearrow C \le L_{\text{out}}(N)\searrow\). Il gap si riduce con \(N\).
  • Radial area corridor: \(A_{\text{in}}(N)\nearrow A \le A_{\text{out}}(N)\searrow\).
  • Cartesian corridor (quarter): \(A_{\text{right,q}}\le A_{\text{mid,q}}\le A_{\text{quarter}}\le A_{\text{trap,q}}\) (arco concavo); disco via \(4\times\).
Classical cross-check (optional, non usato da GRG per calcolare). Con trigonometria classica: \[ \begin{aligned} C &\ge 8\,N\,R\,\sin\!\Bigl(\tfrac{\pi}{4N}\Bigr),\\ C &\le 8\,N\,R\,\tan\!\Bigl(\tfrac{\pi}{4N}\Bigr), \end{aligned} \] da cui un gap \(=\Theta(N^{-2})\).
(Turns equivalence, funzioni a periodo 1) con \(s(u):=\sin(2\pi u)\), \(t(u):=\tan(2\pi u)\): \[ \begin{aligned} C &\ge 8\,N\,R\,s\!\Bigl(\tfrac{1}{8N}\Bigr),\\ C &\le 8\,N\,R\,t\!\Bigl(\tfrac{1}{8N}\Bigr). \end{aligned} \] Stesso ordine \(O(N^{-2})\) per il trapezoide cartesiano su un quarto concavo. Sono controlli esterni, non prerequisiti di GRG.